Chapter 4: Rational, Power, and Root Functions

Chapter 4: Rational, Power, and Root Functions 4.1 Rational Functions and Graphs 4.2 More on Rational Functions and Graphs 4.3 Rational Equations, Inequalities, Models, and Applications 4.4 Functions Defined by Powers and Roots 4.5 Equations, Inequalities, and Applications Involving Root Functions

4.3 Rational Equations, Inequalities, Applications, and Models • Solving Rational Equations and Inequalities • at least one variable in the denominator • may be undefined for certain values where the denominator is 0 • identify those values that make the equation (or inequality) undefined • when solving rational equations, you generally multiply both sides by a common denominator • when solving rational inequalities, you generally get 0 on one side, then rewrite the rational expression as a single fraction

4.3 Solving a Rational Equation Analytically Example Solve Analytic Solution Notice that the expression is undefined for Multiply both sides by 2x + 1. Solve for x. The solution set is {1}.

4.3 Solving a Rational Equation Analytically Graphical Solution Rewrite the equation as and define Y1 Using the x-intercept method shows that the zero of the function is 1.

4.3 Solving a Rational Equation Example Solve Solution For this equation,

4.3 Solving a Rational Equation But, x = 2 is not in the domain of the original equation and, therefore, must be rejected. The solution set is {–5}.

4.3 Solving a Rational Inequality Analytically Example Solve the rational inequality Analytic Solution We can’t multiply both sides by 2x + 1 since it may be negative. Start by subtracting 1 from both sides. Common denominator is 2x + 1.

4.3 Solving a Rational Inequality Analytically Rewrite as a single fraction. To determine the sign graph, solve the equations to get x = 1 and

4.3 Solving a Rational Inequality Analytically Complete the sign graph and determine the intervals where the quotient is negative. The quotient is zero or negative when x is in Can’t include it makes the denominator 0.

4.3 Solving a Rational Inequality Graphically Graphical Solution Let Y1 We use the graph to find the intervals where Y1 is below the x-axis, including the x-intercepts, where Y1 = 0. The solution set is

4.3 Solving Equations Involving Rational Functions • Solving a Rational Equation • Rewrite the inequality, if necessary, so that 0 is on one side and there is a single rational expression on the other side. • Determine the values that will cause either the numerator or the denominator of the rational expression to equal 0. These values determine the intervals on the number line to consider. • Use the test value from each interval to determine which intervals form the solution set. Be sure to check endpoints.

4.3 Models and Applications of Rational Functions: Analyzing Traffic Intensity Example Vehicles arrive randomly at a parking ramp at an average rate of 2.6 vehicles per minute. The parking attendant can admit 3.2 cars per minute. However, since arrivals are random, lines form at various times. • The traffic intensityx is defined as the ratio of the average arrival rate to the average admittance rate. Determine x for this parking ramp. • The average number of vehicles waiting in line to enter the ramp is modeled by f(x) = where 0  x <1 is the traffic intensity. Compute f(x) for this parking ramp. (c) Graph y = f(x). What happens to the number of vehicles waiting as the traffic intensity approaches 1?

4.3 Models and Applications of Rational Functions: Analyzing Traffic Intensity Solution • Average arrival rate = 2.6 vehicles/min, average admittance rate = 3.2 vehicles/min, so (b) From part (a), the average number of vehicles waiting in line is f(.8125).

4.3 Models and Applications of Rational Functions: Analyzing Traffic Intensity (c) From the graph below, we see that as x approaches 1, y = f(x) gets very large, that is, the number of waiting vehicles gets very large.

4.3 Models and Applications of Rational Functions: Optimization Problem Example A manufacturer wants to construct cylindrical aluminum cans with volume 2000 cm3 (2 liters). What radius and height will minimize the amount of aluminum used? What will this amount be? Solution Two unknowns: radius x and height h. To minimize the amount of aluminum, we minimize the surface area. Volume V is

4.3 Models and Applications of Rational Functions: Optimization Problem Surface area S = 2xh + 2x2, x > 0 (since x is the radius), can now be written as a function of x. Minimum radius is approximately 6.83 cm and the height associated with that is 13.65 cm, giving a minimum amount of aluminum of 878.76 cm3.

4.3 Inverse Variation Inverse Variation as the nth Power Let x and y denote two quantities and n be a positive number. Then y is inversely proportional to the nth power of x, or yvaries inversely as the nth power of x, if there exists a nonzero number k such that If then y is inversely proportional to x, or y variesinversely as x.

4.3 Modeling the Intensity of Light The intensity of light I is inversely proportional to the second power of the distance d. The equation models this phenomenon. At a distance of 3 meters, a 100-watt bulb produces an intensity of 0.88 watt per square meter. Find the constant of variation k, and then determine the intensity of the light at a distance of 2 meters. Substitute d = 3, and I = .88 into the variation equation, and solve for k.

4.3 Joint Variation Joint Variation Let m and n be real numbers. Then zvaries jointly as the nth power of x and the mth power of y if a nonzero real number k exists such that z = kxnym.

4.3 Solving a Combined Variation Problem In the photography formula the luminance L (in foot-candles) varies directly as the square of the F-stop F and inversely as the product of the file ASA number s and the shutter speed t. The constant of variation is 25. Suppose we want to use 200 ASA file and a shutter speed of 1/250 when 500 foot candles of light are available. What would be an appropriate F-stop? An F-stop of 4 would be appropriate.

4.3 Rate of Work Rate of Work If 1 task can be completed in x units of time, then the rate of work is 1/x task per time unit.

4.3 Analyzing Work Rate Example It takes machine B one hour less to complete a task when working alone than it takes machine A working alone. If they start together, they can complete the task in 72 minutes. How long does it take each machine to complete the task when working alone? Solution Let x represent the number of hours it takes machine A to complete the task alone. Then it takes machine B hours working alone.

4.3 Analyzing Work Rate Solution The only value that makes sense is 3. It takes machine A 3 hours to complete the task alone, and it takes machine B 2 hours to complete the task alone.

Chapter 4: Rational, Power, and Root Functions